Subordination of Some Cubic Polynomials Associated with Convex Functions

Let \(V_3(z,f)= (3/4)z+(3/10)a_2z^2+(1/20)a_3z^3\) and \(\sigma _3^{(\alpha )}(z,f)=z+(2/(2+\alpha ))a_2z^2+(2/[(2+\alpha )(1+\alpha )])a_3z^3\) be the cubic polynomials representing, respectively, the 3rd de la Vall\(\acute{e}\)e Poussin mean and the 3rd Ces\(\grave{a}\)ro mean of order \(\alpha \; (\alpha \ge 0)\) of a normalized analytic function \(f(z) = z+\sum _{k=2}^{\infty } a_k z^k.\) If \(\mathscr {K}\) denotes the usual class of normalized convex univalent functions in the open unit disc \({\mathbb {D}}\) in the complex plane centered at the origin, we demonstrate that \(V_3(z,f)\prec \sigma _3^{(\alpha )}(z,f)\) in \(\mathbb {D}\) for all \(f\in \mathscr {K}\) and for all real numbers \(\alpha\) satisfying \(3\le \alpha \le 19\). For all \(\alpha \ge 19,\) we also identify a sharp real number (multiplier) \(\gamma (\alpha )\) such that \(\gamma (\alpha )\cdot V_3(z,f)\prec \sigma _3^{(\alpha )}(z,f)\) in \(\mathbb {D}\) for all \(f\in \mathscr {K}.\) Here ‘\(\prec\)’ denotes subordination between two analytic functions.